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arXiv:math-ph/9810017 · International Journal of Theoretical Physics

The Representation Theory of Decoherence Functionals in History Quantum Theories

Oliver Rudolph

Published Mar 1, 2000• 3 citations
arXivHTMLPDF
DOI
10.1023/a:1003635131465

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A general quantum history theory can be characterized by the space of histories and by the space of decoherence functionals. In this note we consider the situation where the space of histories is given by the lattice of projection operators on an infinite dimensional Hilbert space H. We study operator representations for decoherence functionals on this space of histories. We first give necessary and sufficient conditions for a decoherence functional being representable by a trace class operator on H⊗H, an infinite dimensional analogue of the Isham–Linden–Schreckenberg representation for finite dimensions. Since this excludes many decoherence functionals of physical interest, we then identify the large and physically important class of decoherence functionals which can be represented, canonically, by bounded operators on H⊗H.
We consider two technical developments of the formalism of continuous-time histories. First, we provide an explicit description of histories of the simple harmonic oscillator on the classical history phase space, comparing and contrasting the Q, P, and Wigner representations; we conclude that a representation based on coherent states is the most appropriate. Second, we demonstrate a generic method for implementing a perturbative approach for interacting theories in the history formalism, using the quartic anharmonic oscillator. We make use of the identification of the closed-time-path generating functional with the decoherence functional to develop a perturbative expansion for the latter up to second order in the coupling constant. We consider both configuration space and phase space histories.